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<h1 id="CR-Nonconforming-Element-for-Poisson-Equation-in-2D">CR Nonconforming Element for Poisson Equation in 2D<a class="anchor-link" href="#CR-Nonconforming-Element-for-Poisson-Equation-in-2D">&#182;</a></h1>
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<p>This example is to show the rate of convergence of the CR Nonconforming finite element approximation of the Poisson equation on the unit square:</p>
$$- \Delta u = f \; \hbox{in } (0,1)^2$$<p>for the following boundary conditions</p>
<ul>
<li>Non-empty Dirichlet boundary condition: $u=g_D \hbox{ on }\Gamma_D, \nabla u\cdot n=g_N \hbox{ on }\Gamma_N.$</li>
<li>Pure Neumann boundary condition: $\nabla u\cdot n=g_N \hbox{ on } \partial \Omega$.</li>
<li>Robin boundary condition: $g_R u + \nabla u\cdot n=g_N \hbox{ on }\partial \Omega$.</li>
</ul>
<p><strong>References</strong>:</p>
<ul>
<li><a href="femdoc.html">Quick Introduction to Finite Element Methods</a></li>
<li><a href="http://www.math.uci.edu/~chenlong/226/Ch2FEM.pdf">Introduction to Finite Element Methods</a></li>
<li><a href="http://www.math.uci.edu/~chenlong/226/Ch3FEMCode.pdf">Progamming of Finite Element Methods</a></li>
</ul>
<p><strong>Subroutines</strong>:</p>

<pre><code>- PoissonCR
- squarePoisson
- femPoisson
- PoissonCRfemrate

</code></pre>
<p>The method is implemented in <code>PoissonCR</code> subroutine and can be tested in <code>squarePoisson</code>. Together with other elements (P1, P2, P3, Q1, CR), <code>femPoisson</code> provides a concise interface to solve Poisson equation. The CR element is tested in <code>PoissonCRfemrate</code>. This doc is based on <code>PoissonCRfemrate</code>.</p>

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<h2 id="CR-Nonconforming-Element">CR Nonconforming Element<a class="anchor-link" href="#CR-Nonconforming-Element">&#182;</a></h2><p>We explain degree of freedoms and basis functions for Crouzeix-Raviart nonconforming P1 element on triangles. The dofs are associated to edges. Given a mesh, the required data structure can be constructured by</p>

<pre><code>  [elem2edge,edge] = dofedge(elem);

</code></pre>
<h3 id="Local-indexing">Local indexing<a class="anchor-link" href="#Local-indexing">&#182;</a></h3>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Local indexing of DOFs</span>
<span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">];</span>
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">3</span><span class="p">];</span>
<span class="n">edge</span> <span class="p">=</span> <span class="p">[</span><span class="mi">2</span> <span class="mi">3</span><span class="p">;</span> <span class="mi">3</span> <span class="mi">1</span><span class="p">;</span> <span class="mi">1</span> <span class="mi">2</span><span class="p">];</span>
<span class="n">figure</span><span class="p">;</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
<span class="n">findedge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">edge</span><span class="p">);</span>
</pre></div>

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rkJggg==
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<h3 id="A-Local-Basis">A Local Basis<a class="anchor-link" href="#A-Local-Basis">&#182;</a></h3><p>The 3 Lagrange-type bases functions are denoted by $\phi_i, i=1:3$, i.e. $\phi_i(m_j)=\delta _{ij},i,j=1:3$, where $m_i$ is the middle point of the i-th edge. In barycentric coordinates, they are:</p>
$$\phi_i = 1- 2\lambda_i,\quad \nabla \phi_i = -2\nabla \lambda_i,$$<p>When transfer to the reference triangle formed by $(0,0),(1,0),(0,1)$, the local bases in x-y coordinate can be obtained by substituting</p>
$$\lambda _1 = x, \quad \lambda _2 = y, \quad \lambda _3 = 1-x-y.$$<p></p>

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<h3 id="Local-to-global-index-map">Local to global index map<a class="anchor-link" href="#Local-to-global-index-map">&#182;</a></h3><p>The matrix <code>elem2edge</code> is the local to the global index mapping of edges. It can be constructed by</p>

<pre><code>[elem2edge,edge] = dofedge(elem);</code></pre>

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<div class=" highlight hl-matlab"><pre><span></span><span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">];</span>
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">;</span> <span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">];</span>      
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">uniformbisect</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">figure</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span> <span class="n">clf</span><span class="p">;</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
<span class="n">findelem</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="p">[</span><span class="n">elem2edge</span><span class="p">,</span><span class="n">edge</span><span class="p">]</span> <span class="p">=</span> <span class="n">dofedge</span><span class="p">(</span><span class="n">elem</span><span class="p">);</span>
<span class="n">findedge</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">edge</span><span class="p">);</span>
<span class="n">display</span><span class="p">(</span><span class="n">elem2edge</span><span class="p">);</span>
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<pre>
elem2edge =

  8�3 uint32 matrix

    5    6   15
   10   11   14
    2    1   13
    7    9   16
    7   15    8
    2   14    3
    5   13    4
   10   16   12

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<h2 id="Mixed-boundary-condition">Mixed boundary condition<a class="anchor-link" href="#Mixed-boundary-condition">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">%% Setting</span>
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">squaremesh</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span><span class="mf">0.25</span><span class="p">);</span> 
<span class="n">mesh</span> <span class="p">=</span> <span class="n">struct</span><span class="p">(</span><span class="s">&#39;node&#39;</span><span class="p">,</span><span class="n">node</span><span class="p">,</span><span class="s">&#39;elem&#39;</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">option</span><span class="p">.</span><span class="n">L0</span> <span class="p">=</span> <span class="mi">2</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">maxIt</span> <span class="p">=</span> <span class="mi">4</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">printlevel</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="n">option</span><span class="p">.</span><span class="n">elemType</span> <span class="p">=</span> <span class="s">&#39;CR&#39;</span><span class="p">;</span>
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<div class=" highlight hl-matlab"><pre><span></span><span class="c">% Mixed boundary condition</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosdata</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Dirichlet&#39;</span><span class="p">,</span><span class="s">&#39;~(x==0)&#39;</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">,</span><span class="s">&#39;x==0&#39;</span><span class="p">);</span>
<span class="n">femPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     3136,  #nnz:    11040, smoothing: (1,1), iter: 12,   err = 2.67e-09,   time = 0.15 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    12416,  #nnz:    44608, smoothing: (1,1), iter: 12,   err = 2.62e-09,   time = 0.11 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    49408,  #nnz:   179328, smoothing: (1,1), iter: 12,   err = 2.57e-09,   time = 0.19 s

 #Dof       h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

  800   6.250e-02   1.20226e-03   1.62318e-01   3.64423e-02   1.55737e-03
 3136   3.125e-02   3.01351e-04   8.12476e-02   1.81858e-02   3.97664e-04
12416   1.562e-02   7.53872e-05   4.06349e-02   9.08851e-03   1.00099e-04
49408   7.812e-03   1.88499e-05   2.03188e-02   4.54371e-03   2.50778e-05

 #Dof   Assemble     Solve      Error      Mesh    

  800   6.00e-02   8.15e-03   8.00e-02   1.00e-02
 3136   6.00e-02   1.45e-01   3.00e-02   1.00e-02
12416   1.60e-01   1.09e-01   4.00e-02   5.00e-02
49408   2.50e-01   1.90e-01   1.00e-01   1.50e-01


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<h2 id="Pure-Neumann-boundary-condition">Pure Neumann boundary condition<a class="anchor-link" href="#Pure-Neumann-boundary-condition">&#182;</a></h2><p>When pure Neumann boundary condition is posed, i.e., $-\Delta u =f$ in $\Omega$ and $\nabla u\cdot n=g_N$ on $\partial \Omega$, the data should be consisitent in the sense that $\int_{\Omega} f \, dx + \int_{\partial \Omega} g \, ds = 0$. The solution is unique up to a constant. A post-process is applied such that the constraint $\int_{\Omega}u_h dx = 0$ is imposed.</p>

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<div class=" highlight hl-matlab"><pre><span></span><span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosNeumanndata</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Neumann&#39;</span><span class="p">);</span>
<span class="n">femPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     3136,  #nnz:    11325, smoothing: (1,1), iter: 13,   err = 4.57e-09,   time = 0.058 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    12416,  #nnz:    45181, smoothing: (1,1), iter: 14,   err = 1.91e-09,   time = 0.058 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    49408,  #nnz:   180477, smoothing: (1,1), iter: 14,   err = 3.96e-09,   time = 0.16 s

 #Dof       h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

  800   6.250e-02   5.18787e-03   6.47906e-01   1.49052e-01   6.33147e-03
 3136   3.125e-02   1.30793e-03   3.24817e-01   7.31524e-02   1.60037e-03
12416   1.562e-02   3.27672e-04   1.62518e-01   3.64052e-02   4.01216e-04
49408   7.812e-03   8.19609e-05   8.12726e-02   1.81812e-02   1.00375e-04

 #Dof   Assemble     Solve      Error      Mesh    

  800   6.00e-02   6.22e-04   2.00e-02   0.00e+00
 3136   1.00e-02   5.78e-02   2.00e-02   0.00e+00
12416   4.00e-02   5.77e-02   4.00e-02   1.00e-02
49408   1.50e-01   1.65e-01   9.00e-02   6.00e-02


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<h2 id="Robin-boundary-condition">Robin boundary condition<a class="anchor-link" href="#Robin-boundary-condition">&#182;</a></h2>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">option</span><span class="p">.</span><span class="n">plotflag</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">pde</span> <span class="p">=</span> <span class="n">sincosRobindata</span><span class="p">;</span>
<span class="n">mesh</span><span class="p">.</span><span class="n">bdFlag</span> <span class="p">=</span> <span class="n">setboundary</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">,</span><span class="s">&#39;Robin&#39;</span><span class="p">);</span>
<span class="n">femPoisson</span><span class="p">(</span><span class="n">mesh</span><span class="p">,</span><span class="n">pde</span><span class="p">,</span><span class="n">option</span><span class="p">);</span>
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<pre>Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:     3136,  #nnz:    11328, smoothing: (1,1), iter: 12,   err = 1.82e-09,   time = 0.059 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    12416,  #nnz:    45184, smoothing: (1,1), iter: 11,   err = 9.89e-09,   time = 0.056 s
Multigrid V-cycle Preconditioner with Conjugate Gradient Method
#dof:    49408,  #nnz:   180480, smoothing: (1,1), iter: 12,   err = 1.78e-09,   time = 0.19 s

 #Dof       h        ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}

  800   6.250e-02   5.14818e-03   6.47717e-01   1.48743e-01   6.97115e-03
 3136   3.125e-02   1.29781e-03   3.24794e-01   7.31131e-02   1.75333e-03
12416   1.562e-02   3.25127e-04   1.62515e-01   3.64002e-02   4.38609e-04
49408   7.812e-03   8.13241e-05   8.12722e-02   1.81806e-02   1.09622e-04

 #Dof   Assemble     Solve      Error      Mesh    

  800   6.00e-02   6.97e-04   1.00e-02   0.00e+00
 3136   1.00e-02   5.89e-02   1.00e-02   0.00e+00
12416   5.00e-02   5.60e-02   3.00e-02   2.00e-02
49408   1.50e-01   1.92e-01   1.10e-01   6.00e-02


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<h2 id="Conclusion">Conclusion<a class="anchor-link" href="#Conclusion">&#182;</a></h2><p>The optimal rate of convergence of the H1-norm (1st order) and L2-norm (2nd order) is observed. No superconvergence for $\|\nabla u_I - \nabla u_h\|$.</p>
<p>MGCG converges uniformly in all cases.</p>

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